Lee could be on the way back

Carlos Lee is a few days away from beginning a rehab assignment as he nears a return from a hamstring strain.

Lee, who is eligible to come off the disabled list Sunday, ran Tuesday and will run again today. Brad Mills said if that goes well, he’ll send him out for rehab. As for a minimum stay on the DL, Mills couldn’t commit.

“We’re just going to have to see how he goes through all that before we can even think about (a return date),” Mills said.

Lee said most of the pain is gone after a miserable first couple of days with the injury.

Four working toward return

More updates on a stable of injured Astros:

Fernando Martinez is back in Houston seeing specialists for his post-concussion syndrome and could be ready to begin physical activity in the next couple of days to see how he responds.

Travis Buck’s 15-day milestone Thursday will likely go by without his being ready, but he said he’s feeling some relief in his Achilles tendon after a slow healing process. He’s unsure whether he’ll need a rehab assignment.Jordan Schafer was out of the lineup Tuesday as the Astros stacked the top seven spots with righthanded or switch hitters against Madison Bumgarner, who is extremely tough lefty-on-lefty. Schafer is still battling an illness, though, which has dragged on more than a week.

Fernando Abad had a second rehab assignment Tuesday night, and the Astros planned to re-evaluate him afterward to see if he’s ready to return to the big leagues.

Altuve garnering All-Star votes

Jose Altuve moved up one spot in the second release of All-Star vote totals, now sitting fourth among National League second baseman.
Altuve trails Atlanta’s Dan Uggla, who leads with 1,341,088 votes to his 700,640, and also trails Brandon Phillips of the Reds and Omar Infante of the Marlins.

Altuve is the only Astro in the top five (or top 15 in the outfield) at his position.

Five more picks now on board

The Astros announced the signings of five more draft picks, bringing the total list to 26 of their 41 picks, with four of those 41 ineligible to sign because their teams made the College World Series.

Outfielders Brett Phillips (Round 6) and Marc Wik (21), pitchers Kenneth Long (22) and Michael Hauschild (33) and infielder Ryan Dineen (25) all signed. Phillips brings the total to six of the 11 players who will impact the top-10-round bonus pool, with Lance McCullers (supplemental first) and Rio Ruiz (fourth) plus three College World Series participants being the exceptions.

Photo: Thearon W. Henderson / 2012 Getty Images

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June 14: Astros 6, Giants 3

Wandy Rodriguez struck out three batters en route to his sixth win of the season.

June 14: Astros 6, Giants 3

Wandy Rodriguez struck out three batters en route to his sixth win of the season.

Photo: Thearon W. Henderson / 2012 Getty Images

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J.D. Martinez is high-fived in the dugout after his grand slam off pitcher Barry Zito during the third inning.

J.D. Martinez is high-fived in the dugout after his grand slam off pitcher Barry Zito during the third inning.

Photo: Marcio Jose Sanchez / Assocaited Press

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Ryan Theriot gets the put out at second base while leaping over the sliding Brian Bixler in the fifth inning.

Ryan Theriot gets the put out at second base while leaping over the sliding Brian Bixler in the fifth inning.

Photo: Thearon W. Henderson / 2012 Getty Images

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Wandy Rodriguez recorded his sixth win of the season against the Giants. He lasted six inning, allowed six hits and three arned runs.

Wandy Rodriguez recorded his sixth win of the season against the Giants. He lasted six inning, allowed six hits and three arned runs.

Photo: Thearon W. Henderson / 2012 Getty Images

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Gregor Blanco steals second base beating the throw to Jose Altuve in the fifth inning.

Gregor Blanco steals second base beating the throw to Jose Altuve in the fifth inning.

Photo: Thearon W. Henderson / 2012 Getty Images

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J.D. Martinez swings and hits a grand slam in the third inning.

J.D. Martinez swings and hits a grand slam in the third inning.

Photo: Thearon W. Henderson / 2012 Getty Images

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Matt Downs swings and watches the flight of his ball as he hits a solo home run in the second inning.

Matt Downs swings and watches the flight of his ball as he hits a solo home run in the second inning.

Photo: Thearon W. Henderson / 2012 Getty Images

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Matt Downs rounds the bases and is congratulated by third base coach Dave Clark after hitting a solo home run in the second inning.

Matt Downs rounds the bases and is congratulated by third base coach Dave Clark after hitting a solo home run in the second inning.

Photo: Thearon W. Henderson / 2012 Getty Images

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June 13: Giants 10, Astros 0

Giants pitcher Matt Cain tossed the 22nd perfect game in major league history, striking out a career-high 14 batters on 125 pitches. It was also the first perfect game in Giants franchise history. less

June 13: Giants 10, Astros 0

Giants pitcher Matt Cain tossed the 22nd perfect game in major league history, striking out a career-high 14 batters on 125 pitches. It was also the first perfect game in Giants ... more

Photo: Jeff Chiu / Associated Press

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Giants pitcher Matt Cain, left, celebrates with catcher Buster Posey after the final out of the ninth inningl. Cain pitched the 22nd perfect game in major league history and first for the Giants, striking out a career-high 14 and getting help from two spectacular catches to beat the Houston Astros 10-0. less

Giants pitcher Matt Cain, left, celebrates with catcher Buster Posey after the final out of the ninth inningl. Cain pitched the 22nd perfect game in major league history and first for the Giants, striking out ... more

Photo: Jeff Chiu / Associated Press

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Giants pitcher Matt Cain, left, celebrates with catcher Buster Posey, center, and first baseman Brandon Belt after the final out of the ninth inning.

Giants pitcher Matt Cain, left, celebrates with catcher Buster Posey, center, and first baseman Brandon Belt after the final out of the ninth inning.

Photo: Jeff Chiu / Associated Press

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Giants pitcher Matt Cain celebrates after the final out of the ninth inning. Cain pitched the 22nd perfect game in major league history and first for the Giants, striking out a career-high 14 and getting help from two spectacular catches. less

Giants pitcher Matt Cain celebrates after the final out of the ninth inning. Cain pitched the 22nd perfect game in major league history and first for the Giants, striking out a career-high 14 and getting help ... more

Photo: Jeff Chiu / Associated Press

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Giants pitcher Matt Cain celebrates after the final out of the ninth inning. Cain pitched the 22nd perfect game in major league history and first for the Giants, striking out a career-high 14 and getting help from two spectacular catches. less

Giants pitcher Matt Cain celebrates after the final out of the ninth inning. Cain pitched the 22nd perfect game in major league history and first for the Giants, striking out a career-high 14 and getting help ... more

Photo: Jeff Chiu / Associated Press

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Giants pitcher Matt Cain (18) celebrates after pitching a perfect game against the Astros.

Giants pitcher Matt Cain (18) celebrates after pitching a perfect game against the Astros.